Article
| Title: | A survey of some recent results on Clifford algebras in $\mathbb {R}^4$ (English) |
| Author: | Janovská, Drahoslava |
| Author: | Opfer, Gerhard |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 68 |
| Issue: | 5 |
| Year: | 2023 |
| Pages: | 571-592 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We will study applications of numerical methods in Clifford algebras in $\mathbb {R}^4$, in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in $\mathbb {R}^4$. In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients can be placed on both sides of the powers. Then there are even five different classes of zeros. All results can be extended to other noncommutative algebras in $\mathbb {R}^4$. In the paper by R. Lauterbach and G. Opfer (2014), the authors constructed an exact Jacobi matrix for functions defined in noncommutative algebraic systems without the use of any partial derivative. We applied this technique to find the eigenvalues of the companion matrix as zeros of the companion polynomial by Newton's method. (English) |
| Keyword: | linear equations in quaternions and coquaternions |
| Keyword: | polynomials over $\mathbb {R}^4$ algebras |
| Keyword: | the algebraic eigenvalue problem over noncommutative algebras |
| Keyword: | Newton's method |
| Keyword: | companion matrix and companion polynomial |
| Keyword: | Niven's algorithm |
| MSC: | 15A06 |
| MSC: | 15A18 |
| MSC: | 15A66 |
| DOI: | 10.21136/AM.2023.0182-22 |
| . | |
| Date available: | 2023-10-05T15:09:49Z |
| Last updated: | 2023-10-09 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151834 |
| . | |
| Reference: | [1] Aramanovitch, L. I.: Quaternion non-linear filter for estimation of rotating body attitude.Math. Methods Appl. Sci. 18 (1995), 1239-1255. Zbl 0841.93060, MR 1362940, 10.1002/mma.1670181504 |
| Reference: | [2] Brenner, J. L.: Matrices of quaternions.Pac. J. Math. 1 (1951), 329-335. Zbl 0043.01402, MR 0043761, 10.2140/pjm.1951.1.329 |
| Reference: | [3] Brody, D. C., Graefe, E.-M.: On complexified mechanics and coquaternions.J. Phys. A, Math. Theor. 44 (2011), Article ID 072001, 9 pages. Zbl 1208.81073, MR 2769792, 10.1088/1751-8113/44/7/072001 |
| Reference: | [4] Cockle, J.: On a new imaginary in algebra.Phil. Mag. (3) 34 (1849), 37-47. 10.1080/14786444908646169 |
| Reference: | [5] Cockle, J.: On systems of algebra involving more than one imaginary and on equations of the fifth degree.Phil. Mag. (3) 35 (1849), 434-437. 10.1080/14786444908646384 |
| Reference: | [6] Cocle, J.: On the symbols of algebra, and on the theory of Tessarines.Phil. Mag. (3) 34 (1849), 406-410. 10.1080/14786444908646257 |
| Reference: | [7] Eilenberg, S., Niven, I.: The "fundamental theorem of algebra" for quaternions.Bull. Am. Math. Soc. 50 (1944), 246-248. Zbl 0063.01228, MR 0009588, 10.1090/S0002-9904-1944-08125-1 |
| Reference: | [8] Frenkel, I., Libine, M.: Split quaternionic analysis and separation of the series for $SL(2,\Bbb{R})$ and $SL(2,\Bbb{C})/SL(2,\Bbb{R})$.Adv. Math. 228 (2011), 678-763. Zbl 1264.30037, MR 2822209, 10.1016/j.aim.2011.06.001 |
| Reference: | [9] Gordon, B., Motzkin, T. S.: On the zeros of polynomials over division rings.Trans. Am. Math. Soc. 116 (1965), 218-226. Zbl 0141.03002, MR 0195853, 10.1090/S0002-9947-1965-0195853-2 |
| Reference: | [10] Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers.Mathematical Methods in Practice. Wiley, Chichester (1997). Zbl 0897.30023 |
| Reference: | [11] Horn, R. A., Johnson, C. R.: Topics in Matrix Analysis.Cambridge University Press, Cambridge (1991). Zbl 0729.15001, MR 1091716, 10.1017/CBO9780511840371 |
| Reference: | [12] Janovská, D., Opfer, G.: Givens' transformation applied to quaternion valued vectors.BIT 43 (2003), 991-1002. Zbl 1052.65030, MR 2058880, 10.1023/B:BITN.0000014561.58141.2c |
| Reference: | [13] Janovská, D., Opfer, G.: Linear equations in quaternionic variables.Mitt. Math. Ges. Hamb. 27 (2008), 223-234. Zbl 1179.11042, MR 2528369 |
| Reference: | [14] Janovská, D., Opfer, G.: A note on the computation of all zeros of simple quaternionic polynomials.SIAM J. Numer. Anal. 48 (2010), 244-256. Zbl 1247.65060, MR 2608368, 10.1137/090748871 |
| Reference: | [15] Janovská, D., Opfer, G.: The classification and the computation of the zeros of quaternionic, two-sided polynomials.Numer. Math. 115 (2010), 81-100. Zbl 1190.65075, MR 2594342, 10.1007/s00211-009-0274-y |
| Reference: | [16] Janovská, D., Opfer, G.: Linear equations and the Kronecker product in coquaternions.Mitt. Math. Ges. Hamb. 33 (2013), 181-196. Zbl 1298.15006, MR 3157447 |
| Reference: | [17] Janovská, D., Opfer, G.: The number of zeros of unilateral polynomials over coquaternions and related algebras.ETNA, Electron. Trans. Numer. Anal. 46 (2017), 55-70. Zbl 1368.65069, MR 3622882 |
| Reference: | [18] Janovská, D., Opfer, G.: The relation between the companion matrix and the companion polynomial in $\Bbb{R}^4$ algebras.Adv. Appl. Clifford Algebr. 28 (2018), Article ID 76, 16 pages. Zbl 1431.15016, MR 3832106, 10.1007/s00006-018-0892-5 |
| Reference: | [19] Johnson, R. E.: On the equation $\chi\alpha=\gamma\chi+\beta$ over an algebraic division ring.Bull. Am. Math. Soc. 50 (1944), 202-207. Zbl 0061.05505, MR 0009947, 10.1090/S0002-9904-1944-08112-3 |
| Reference: | [20] Lam, T. Y.: A First Course in Noncommutative Rings.Graduate Texts in Mathematics 131. Springer, New York (1991). Zbl 0728.16001, MR 1125071, 10.1007/978-1-4684-0406-7 |
| Reference: | [21] Lauterbach, R., Opfer, G.: The Jacobi matrix for functions in noncommutative algebras.Adv. Appl. Clifford Algebr. 24 (2014), 1059-1073. Zbl 1316.65049, MR 3277688, 10.1007/s00006-014-0504-y |
| Reference: | [22] Lee, H. C.: Eigenvalues of canonical forms of matrices with quaternion coefficients.Proc. Irish Acad. A 52 (1949), 253-260. Zbl 0036.29802, MR 0036738 |
| Reference: | [23] Niven, I.: Equations in quaternions.Am. Math. Mon. 48 (1941), 654-661. Zbl 0060.08002, MR 0006159, 10.1080/00029890.1941.11991158 |
| Reference: | [24] Ore, Ø.: Linear equations in non-commutative fields.Ann. Math. (2) 32 (1931), 463-477. Zbl 0001.26601, MR 1503010, 10.2307/1968245 |
| Reference: | [25] Ore, Ø.: Theory of non-commutative polynomials.Ann. Math. (2) 34 (1933), 480-508. Zbl 0007.15101, MR 1503119, 10.2307/1968173 |
| Reference: | [26] Pogorui, A., Shapiro, M.: On the structure of the set of zeros of quaternionic polynomials.Complex Variables, Theory Appl. 49 (2004), 379-389. Zbl 1160.30353, MR 2073169, 10.1080/0278107042000220276 |
| Reference: | [27] Poodiack, R. D., LeClair, K. J.: Fundamental theorems of algebra for the perplexes.College Math. J. 40 (2009), 322-335. MR 2572201, 10.4169/074683409X475643 |
| Reference: | [28] Schmeikal, B.: Tessarinen, Nektarinen und andere Vierheiten: Beweis einer Beobachtung von Gerhard Opfer.Mitt. Math. Ges. Hamb. 34 (2014), 81-108 German. Zbl 1310.15038, MR 3309616 |
| Reference: | [29] Waerden, B. L. van der: Algebra.Die Grundlehren der mathematischen Wissenschaften 33. Springer, Berlin (1960), German. Zbl 0087.25903, MR 0122834, 10.1007/978-3-662-01380-9 |
| Reference: | [30] Wolf, L. A.: Similarity of matrices in which the elements are real quaternions.Bull. Am. Math. Soc. 42 (1936), 737-743. Zbl 0015.24206, MR 1563415, 10.1090/S0002-9904-1936-06417-7 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
